The price competition à-la Bertrand is modeled classically by a Nash game. There are several results concerning existence and uniqueness of the corresponding Nash equilibria. For linear random utility models with an \(-1/(n+1)\) -concave density function and each of the \(n\)  firms offering exactly one product, a price equilibrium exists [1]}. In the symmetric case where additionally all observable utilities of alternative are equal, the result can be strengthened to a unique Nash equilibrium [1]}. With slightly more assumptions, the same authors show the existence of a subgame perfect Nash equilibrium under nested logit demand. In [3]} the multiproduct pricing problem is reduced to finding the root of a single valued equation. The equilibrium price of such an oligopolistic competition is determined by using a modified Lambert W function. The idea of dimensionality reduction can be found in [4]}, where also the existence of a unique Nash equilibrium is proven.
